An implicit BDF2 dual-time-stepping scheme produced practically the same cylinder accuracy as the explicit SSPRK3 method and nearly identical Taylor–Green vortex diagnostic histories in two numerical high-Mach flow benchmarks, according to an arXiv preprint. In the three-dimensional Taylor–Green test, the paper reports 11.4% fewer steps and 27% lower average wall-clock time for BDF2 DTS than for SSPRK3. The document is an arXiv v1 preprint dated 20 Aug 2026.
How the scheme works
Known as BDF2 DTS, the proposed method is an implicit, second-order scheme for the three-dimensional compressible Navier–Stokes equations. It uses high-order spectral collocation and flux limiting to combine a first-order positivity-preserving scheme with a high-order scheme that can violate positivity on each element.
Positivity here concerns keeping the computed density and internal energy positive. The paper’s density proof requires positive density at the relevant step and a stated constraint; it also says that, if the previous pseudostep is admissible, a positive pseudotime increment exists that preserves both density and internal-energy positivity.
In its conclusion, the paper describes the scheme as combining implicit-BDF2 unconditional-stability properties with positivity and entropy stability, design-order physical-time accuracy, and no physical-time-step constraint. The numerical evidence in the paper, however, comes from the two benchmark settings described below.
Two benchmark flows
The first benchmark was a two-dimensional hypersonic cylinder flow at Mach 17.605 and Reynolds number 376,930. It used 55,216 elements at polynomial order 5. BDF2 DTS and SSPRK3 were reported to have practically identical pressure-coefficient results and practically the same skin-friction accuracy, while the BDF2 physical timestep averaged 20 times higher.
The second was a three-dimensional Taylor–Green vortex flow at Mach 10 and Reynolds number 400, using 643 elements at polynomial order 6. On that grid, BDF2 DTS and SSPRK3 produced nearly identical histories for total kinetic energy and dilational kinetic-energy dissipation. In the compared pressure and x-velocity profiles, BDF2 DTS was slightly more dissipative.
Where the timing gain appeared
In the Taylor–Green efficiency run, BDF2 DTS used 5,249 reported explicit steps, compared with 5,925 for SSPRK3—an 11.4% reduction reported by the authors. Average wall-clock times over three runs were 48,619 seconds for BDF2 DTS and 66,683 seconds for SSPRK3; the paper reports a 27% reduction for BDF2 DTS.
An additional comparison with SSPRK2 recorded 3,868 explicit steps and a three-run average wall-clock time of 51,648 seconds. The paper reports SSPRK2’s wall-clock time as 5.8% higher than BDF2 DTS, but it did not assess SSPRK2 accuracy.
What the tests do not settle
Those figures do not amount to a broad validation. The study tests the cylinder and Taylor–Green configurations only; for the Mach 10 Taylor–Green case, the authors say no reference solution or experimental data were available. Agreement with SSPRK3 in that case is therefore a comparison with the paper’s numerical comparator, not an independent check.
Runtime is tied to the test setup as well: the wall-clock figures were averaged over three runs on a fixed hardware configuration, and the paper notes that efficiency can depend on hardware and implementation. The reported reduction therefore does not establish that the same saving will appear in other environments.
Paper data and sources
Original title: Implicit BDF2 dual time-stepping positivity-preserving entropy-stable schemes for unsteady compressible viscous flows
Authors: Mohammed Sayyari, Nail K. Yamaleev
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text